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Periodic Signals

Fourier series for periodic signals

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Even Signals

Odd Signals

Odd square wave

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Even square wave

LF Generator + Oscilloscope

VR2 =R2

R1 + R2V = R2 = 1M Ω >> R1 = 50 Ω = V ;

f0 = 99.9981 kHz

Vpp = 2.04 V

It is set on the LF generator Vout = 1 V

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LF Generator + SPA

VR4 =R4

R3 + R4V =

1

2V ;

Link between OX and SPA• The SPA measures the signal strength at 50 ohms. To evaluate this power when 1 V is set

on the LF generator we must make the following considerations.

• The voltage read on the oscilloscope is the peak-to-peak voltage (𝑉𝑝𝑝). In order to calculate the power, the peak voltage (𝑉𝑝=𝑉𝑝𝑝/2) and its effective value (𝑉𝑒𝑓𝑓=𝑉𝑝/ 2) must be calculated. As noted above it will be necessary to divide again by 2 due to partition losses. So the final power calculation will be:

• Vp =𝑉𝑝𝑝

2= 1.020 V

• VRMS =Vp

2= 0.721 V

• Vx =VRMS

2= 0.361 V

• P =𝑉𝑥

2

50= 0.00260 W = 2.6 mW = 4.14 dBm

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Amplitude and frequencymodulation

Amplitude Modulation

Ac: carrier amplitudefc: carrier frequencya: modulation index (0≤a≤1)m(t): normalised modulating signalAc[1+m(t)]: envelope amplitude carrier Lateral

Bands

V( f )Vc( f )Vs( f )

V(t) = AC [1 + a m(t)] cos(2 fCt + )

V(t) = AC cos(2 fCt ) + AC a m(t) cos(2 fCt ) = vc(t) + vs(t)

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Amplitude ModulationSinusoidal modulation

v(t) Ac cos(2fct) Acacos 2fmt cos(2fct)

v(t) Ac cos(2fct)Aca

2cos 2 fc fm t cos(2 fc fm t)

In addition to the carrier we have twoside bands centered on the sum andthe difference frequencies betweenthat of the carrier and of themodulating signal. Both have anamplitude equal to that of the carrierfor half the modulation inde x

p

SB

A

Aa

2

)2

log(20)()(a

dBAdBA pSB

2

cp

AA

fc fm

fc fm

fc

4

cSB

AA

Amplitude Modulation

Ac: carrier amplitudefc: carrier frequency

a: modulation index (0≤a≤1)m(t): modulating signal

V(t) = AC [1 + a m(t)] cos(2 fCt + )

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Modulation Index measurement

Time Domain Visualizationfc=100 MHzfm=1 kHz

SETTINGCentr. Freq. 100 MHzSPAN 0RBW > BWRANGE LINEARVIDEO TRIGGER

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Frequency Modulation

A: carrier amplitudefc: carrier frequency

m: modulation index

Signal Bandwidth

Max frequency deviation

Frequency Modulation

A: carrier amplitudefc: carrier frequency

m: modulation index

Carson formula

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Frequency Modulation

Frequency modulation on the SPA

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Frequency Modulation with Superimposed AM Modulation

Step recovery Diodes

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Step recovery Diodes• Step recovery diodes are p-i-n

diodes, where the intrinsic region is designed to have an average life time of the charges before recombination as long as possible.

.

Direct Polarization

Reverse Polarization

Step Recovery Diodes• When the diode is biased in direct electrons and holes enter the intrinsic region (accumulation) there is

therefore a direct current in the diode.

• When the external voltage is reversed, the current is also reversed. The intrinsic region begins to empty (emptying) but, due to the presence of the accumulated charges, the voltage at the ends of the diode initially remains at the value that it had in direct.

• When all the charges present in the intrinsic region have been eliminated, the voltage at the ends of the diode is suddenly reversed and the diode is found in reverse polarization (transition and inversion).

• Since the switching speed is quite high, the diode is able to generate a rather steep voltage front, or a signal with a high bandwidth occupation.

RS=50

RL=50

VS

VOUT

t=0 t=t1

Tt

t

t

t

VD(t)

VS(t)Vs

VD

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Transition Times• The transition time, ie the time required to reverse the voltage

across the diode, is related to the frequency band of the diode.

• Considering only the diode, limitations to the response speed are linked to the parasitic elements of the diode, that is to the junction capacity and to the resistance of the semiconductor.

• Therefore, by representing the diode by means of a capacitance and a resistance, the introduced time constant is given by =RC

RC2

1

2

1≤f

=

The diode cut-off frequencies are of the order of 300 - 350 GHz.

cut-off Frequency

Device